On local preservation of orthogonality and its application to isometries

TL;DR

This paper explores how linear operators preserve orthogonality locally in finite-dimensional Banach spaces and applies these insights to refine the characterization of isometries on certain polyhedral spaces.

Contribution

It introduces a study of local orthogonality preservation and connects it to the geometry of Banach spaces, leading to a refinement of existing isometry characterizations.

Findings

Local orthogonality preservation relates to k-smoothness and extremal properties.

Refinement of the Blanco-Koldobsky-Turnsek characterization of isometries.

Application to polyhedral Banach spaces like ℓ∞^n and ℓ₁^n.

Abstract

We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of a Banach space. As an application of the results obtained in this direction, we obtain a refinement of the well-known Blanco-Koldobsky-Turnsek characterization of isometries on some polyhedral Banach spaces, including $ \ell_{\infty}^n, \ell_1^n. $